UNIVERSITY OF VIENNA

WORKING PAPERS Beliefs and Consumer Search Maarten Janssen Sandro Shelegia January 2015 Working Paper No: 1501 DEPARTMENT OF ECONOMICS UNIVERSITY OF VIENNA All our working papers are available at: http://mailbox.univie.ac.at/papers.econ

Beliefs and Consumer Search Maarten Janssen and Sandro Shelegia January 12, 2015 Abstract When consumers search sequentially for prices and product matches, their beliefs of what they will encounter at the next firm are important in deciding whether or not to continue to search. In search environments where retailers have a common cost that is not known to consumers and is either the outcome of a random process or strategically set by an upstream firm, it is natural for consumers to have symmetric beliefs. We show that market outcomes under symmetric beliefs are quantitatively and qualitatively different from outcomes when consumers hold passive beliefs. Market prices are higher with symmetric beliefs (and can be as high as the joint profit maximizing prices), and are non-monotonic in the search cost. Moreover, price rigidities arise endogenously as retailers are not willing to charge prices above consumers reservation utility. These phenomena become exacerbated in a vertical relations environment. JEL Classification: D40; D83; L13 Keywords: Vertical Relations, Consumer Search, Double Marginalization, Product Differentiation, Price Rigidities We have benefitted from comments by Doh-Shin Jeon, José-Luis Moraga-González, Alexei Parakhonyak, Patrick Rey, Andrew Rhodes, Anton Sobolev, Chris Wilson and participants to the IV workshop on Consumer Search (Moscow) and seminars in Barcelona, Milan and Paris, Toulouse, and XXIX Jornadas de Economia Industrial 2014. Department of Economics, University of Vienna and State University Higher School of Economics. Email: maarten.janssen@univie.ac.at Department of Economics, University of Vienna. Email: sandro.shelegia@univie.ac.at 1

1 Introduction In consumer search markets, the market power of firms depends on consumers willingness to search. When at a firm, consumers compare the benefits of buying now with the expected benefits of continuing to search. These expected benefits of search crucially depend on the price consumers believe a next firm charges. Consumers will form beliefs differently depending on the environment they are in and the amount of information (about the environment) they possess. Resulting beliefs determine how consumers react to price changes, and thus how profitable price changes are. If consumers are pessimistic about whether the next search will yield a good offer, they are more likely to accept the current offer, giving firms incentives to set higher prices. This basic insight is important in any search market, but - as we will argue in this paper - it is particularly important in markets where consumers face uncertainty concerning factors that influence market prices. If, for example, consumers know that prices may depend on firms common marginal cost, but are uninformed about this cost, then the price the consumer observes at the current firm is likely to be interpreted as a signal of the common cost level and, thus (presuming prices positively depend on cost), of the expected benefits of search. In a similar way, if retailers common marginal cost is determined by a manufacturer who strategically determines the wholesale price (as an important component in the retailers cost) and consumers do not observe the wholesale price, then beliefs about retail prices that are not yet observed will be influenced by the prices consumers observe at the retailer they are currently visiting. When market power depends on consumer beliefs, it is important to understand their impact of beliefs and what type of beliefs consumers may have. The consumer search literature has, by and large, side-stepped the issue of consumer beliefs and implicitly or explicitly assumed that consumers hold passive beliefs, i.e., no matter what they observe at a particular firm, consumers believe that all other firms will stick to their equilibrium strategies. 1 We have argued above that there are natural environments where one would expect consumers to have other types of beliefs, such as symmetric beliefs, 2 where consumers believe that firms whose prices they have not yet observed charge the same price as the firm they are currently visiting. Going one step further, one may argue that even if consumers fully know the firms environment, consumers may hold symmetric beliefs, i.e., there is no particular reason why consumers should hold passive beliefs even in standard models of consumer search. 1 This is the case in both large strands of the sequential search literature. For heterogeneous goods models that build on Wolinsky (1986), passive beliefs are the norm (e.g. see Anderson and Renault (1999), Armstrong, Vickers and Zhou (2009), and Bar-Isaac, Caruana and Cuñat (2011) among many others). The same is true for homogenous goods models that build on Stahl (1989), but there passive beliefs are necessary for equilibrium to exist (see also Footnote 4). 2 See, for instance, Hart et al. (1990), McAfee and Schwartz (1994) and more recently Pagnozzi and Piccolo (2012). 2

This paper shows that symmetric beliefs lead to equilibrium predictions that are qualitatively and quantitatively significantly different from the traditional analysis that assumes passive beliefs. In particular, we provide new explanations for such diverse phenomena as price rigidities, sales and seemingly collusive outcomes. We do so in three interconnected models. We start our analysis by introducing symmetric beliefs in the standard model by Wolinsky (1986). We then consider the two adaptations of this model that were alluded to in the first paragraph: one where there is asymmetric information about firms common cost and one where this cost is determined by an upstream manufacturer. To explain how these phenomena arise under symmetric beliefs in these different environments, it is important to understand how the Wolinsky model relates to the Diamond Paradox (Diamond (1971)). Diamond showed that with homogenous goods, for any positive search cost, there will be no search beyond the first firm and all firms charge the monopoly price. As discussed in Anderson and Renault (1999), Wolinsky solved the paradox by introducing product differentiation, giving some consumers incentives to search. However, when the search cost is sufficiently high, Wolinsky s solution fails because even consumers with very bad utility draws are not willing to pay the search cost. As a result, when the search cost exceeds a certain threshold, search stops, firms set the monopoly price, and demand drops discontinuously. The existence of such a search cost threshold, and the associated demand drop is also new to the literature. An immediate implication of symmetric beliefs in the Wolinsky (1986) model is that, whenever consumers search in equilibrium, prices are higher with symmetric beliefs. The reason is that with symmetric beliefs consumers are less willing to search after observing a price that is higher than the equilibrium price, so a switch from passive to symmetric beliefs increases prices. A less immediate change, but one that is even more important in the two applications we consider, is that with symmetric beliefs the equilibrium price is non-monotonic in search cost. For relatively small search costs, the price is increasing in search cost (as it is under passive beliefs). Once the search cost is sufficiently high, however, firms find themselves in a situation where, conditional on consumers searching, they would like to charge prices in excess of the consumers reservation utility, but when they do so, consumers do not want to search. The latter increases firms demand elasticity and induces them to reduce prices to the reservation utility. This creates an interval of intermediate search costs where firms charge prices equal to the reservation utility. As the reservation utility is decreasing in search cost, prices are also decreasing. 3 Overall, price is hump-shaped in search cost. Interestingly, at the maximum, both firms set the price that a monopolist selling both goods would set. Thus, with symmetric beliefs and intermediate 3 This inverse relationship between search cost and prices is unlike the earlier contributions with the same conclusion. In Janssen, Moraga-González and Wildenbeest (2005), the search cost changes the composition of heterogeneous consumers and may result in lower prices at higher search cost. In Zhou (2014) consumers search for multiple products. In this environment, products are search complements and if the search cost increases, firms may compete more intensely to prevent consumers searching further. 3

search costs a fully collusive outcome results even though firms act non-cooperatively! Another way to look at the same equilibrium is to fix the search cost, and see how equilibrium changes as the marginal cost increases. This perspective is important for both of our applications where marginal cost plays the key role. With passive beliefs, price is always increasing in marginal cost. With symmetric beliefs, there is an intermediate level of cost where price equals the reservation utility, and because the reservation utility does not depend on marginal cost, neither does price. So with symmetric beliefs the price is first increasing, then constant, and then increasing again. This creates a new search theoretic explanation for price rigidities: firms will not increase prices above consumers reservation utility. We next adapt the Wolinsky model to an environment where firms common marginal cost is random, known to firms but not to consumers. 4 This is relevant in a variety of product markets where the cost of inputs fluctuates, such as products where precious metals and fossil fuels are used. In a separating equilibrium of this model, consumers have to infer marginal cost from the prices they observe, resulting in symmetric beliefs. Once we focus on symmetric beliefs, several conclusions follow immediately from our prior analysis. First, for any sufficiently low realization of marginal cost, prices are higher in the model with uncertain cost than with known cost. Second, for some cost realizations price is equal to reservation utility, and so independent of cost. This allows us to provide an alternative explanation for (i) price stickiness (see Sheshinski and Weiss (1977) and Akerlof and Yellen (1985); Akerlof and L.Yellen (1985) for seminal contributions), and (ii) for the observation that retailers often follow a pricing strategy that is characterized by a regular price and periodic price reductions. It is important to note that our search theoretic explanation of price rigidity does not depend on some exogenously assumed menu cost, nor does it assume that costs are somehow sticky (as in Cabral and Fishman (2012)): cost fluctuates, but for a (large) range of cost realizations, retailers do not change their price. 5 Our explanation of irregular sales is, unlike the existing literature, in terms of a pure strategy equilibrium where a firm s price is a function of its fluctuating cost, but for a range of high cost realizations firms charge the same regular price. Our second application is a vertical relations model where retailers common cost is set by an upstream monopolist and the cost is not observed by consumers. In this environment consumers may have passive or symmetric beliefs as the final retail price 4 In the context of an alternative, homogeneous goods search model by Stahl (1989), Dana (1994), and more recently, Tappata (2009), Chandra and Tappata (2011) and Janssen, Pichler and Weidenholzer (2011) have analyzed consequences of asymmetric information about firms common marginal cost. Janssen and Shelegia (2014) analyze a vertical relations model were retailers common cost is determined by an upstream firm and is unobserved by consumers. These models use passive beliefs, however, as a reservation price equilibrium does not exist under symmetric beliefs in homogeneous goods search models. 5 Cabral and Fishman offer a search theoretic explanation for why output prices are stickier than input costs. In our model, costs are not sticky at all, but prices are. 4

a consumer observes is the product of the wholesale price set by the manufacturer and the way the retailer reacts to that price. Under passive beliefs, the consumer blames an individual retailer for setting an unexpected price and believes that the other firms (the manufacturer and the second retailer) have not deviated. Under symmetric beliefs, the consumer blames the manufacturer for the fact that the consumer has observed an unexpected price and believes that retailers react in the same way to the deviation by the manufacturer. Under passive beliefs, the equilibrium structure is as follows. For low search cost, the manufacturer sets a wholesale price such that retail prices are lower than the reservation utility. For higher search costs, the manufacturer s optimal price is such that retail prices will be higher than the reservation utility and consumers do not search. Thus, for high search cost there is no active search and a classic double-marginalization outcome results, with the retail prices being equal to the retail monopoly price (given the wholesale price) and the manufacturer setting his optimal price (given this retail behavior). Also in the vertical relations model, symmetric beliefs create a difference in that there may exist a large intermediate region of search costs where both wholesale and retail prices are decreasing in search cost. All the features that we have emphasized above, such as price rigidities, non-monotonicity of retail prices in search cost, sales-type pricing behavior and retail prices being higher than monopoly prices (here: double marginalization prices), may also occur in the vertical relations model. The result on price rigidities holds, even if the search cost is very small. In a specific example, we show that the vertical relations model may exacerbate all these phenomena as the manufacturer deliberately sets a wholesale price so as to have retail prices equal to the reservation utility in order to prevent a drop in market demand. To the best of our knowledge, there is no other literature where a firm proactively seeks to prevent the Diamond Paradox from arising. Under symmetric beliefs, the manufacturer is able to manipulate consumers beliefs and, thus, their search behavior. The rest of the paper is organized as follows. In Section 2 we set up the Wolinsky model, and show how a firm s demand depends on consumer beliefs. In Section 3 we analyze firm pricing with passive and symmetric beliefs. In Section 4 we introduce marginal cost uncertainty into the Wolinsky model and solve for equilibrium using our results from previous sections. In Section 5, we introduce an upstream monopolist who determines the retailers marginal cost in the Wolinsky model. We show that by acting non-cooperatively the Diamond Paradox can be avoided by the manufacturer. The final section concludes. Proofs are contained in Appendix A. Subsequent appendices deal with possible alternative formulations of the model. 5

2 The Wolinsky Model and Preliminaries The retail side of the models we study follows Wolinsky (1986). There are two retailers, 1 and 2, who can acquire some input at a common cost c. The retailers transform the input into a final differentiated good, using a one for one technology. There is a unit mass of consumers per retailer. Utility to a consumer from buying the good at retailer i is v i. This utility is drawn from the distribution function G(v), which is the same for both retailers and defined over the (possibly infinite) interval [v, v]. As is standard in the literature, we require that 1 G(v) is logconcave, but in the vertical model, we sometimes restrict ourselves to G(v) being uniformly distributed. The a consumer s valuation of the product of retailer 1 is independent of his valuation of the product of retailer 2. Each consumer visits one of the retailers at random and finds out v i and p i. After observing the match value v i and the price p i consumers decide whether or not to visit the second firm. If they do so, they make their purchase at the best available surplus v i p i, provided that it exceeds zero, the outside option. We assume that the first visit is free. 6 The second visit is costly, and the cost is denoted by s. If after visiting the first retailer, consumers decide to search the second firm, they can always go back to the first retailer at no additional cost (free recall). The above set-up is common. We now explicitly specify the role of beliefs in the model, and derive retailers demand. Assume retailer i sets price p i and retailer j sets price p j. Consider consumers who make their first visit at retailer i, where they find out p i and v i. These consumers do not know p j, and they will have to form some belief about it, denoted by p e j. We write p e j(p i ) as in general, p e j may depend on p i. The same is true for consumers who visit firm j first and have to form beliefs about p i, denoted by p e i (p j ). The consumer search literature implicitly or explicitly uses passive beliefs, implying that p e j is independent of p i (and p e i is independent of p j ). The vertical contracting literature (see, e.g., Hart et al. (1990) and McAfee and Schwartz (1994)) also consider symmetric beliefs, which in our setting can be described as p e j = p i : consumers who first visit firm i believe that firm j charges the same price as i. Many other beliefs are possible, but we shall concentrate on passive and symmetric beliefs for two reasons. First, both have been used in the literature, are well understood and are likely to be used by consumers. Second, once we understand how the standard model changes with symmetric beliefs, it will be relatively straightforward to extrapolate how the equilibrium predictions change with other beliefs. Consider a consumer who visits firm i. As is well known, a rational consumer will use a reservation utility strategy, where she searches if the utility drawn is below a certain threshold, and stops if the utility exceeds this threshold. This threshold depends on the utility realization, firm i s price and the belief about firm j s price. Before finding it, 6 See Appendix C for the solution of our model when the first search is costly. 6

we determine the reservation utility w, a utility level at which a consumer is indifferent between searching the second firm, or accepting w, assuming that both firms charge equal prices. The reservation utility w is the solution to v w (v w)f(v) dv = s, (1) if the solution exists, and is equal to w = v if it does not. The threshold utility level is then simply computed as w plus the (expected) price difference. Formally, a consumer who draws v i and p i will search the other firm if v i < w + (p i p e j). This condition guarantees that the consumer prefers to search rather than to accept the current offer. In order to guarantee that consumers search we need that their reservation utility exceeds the price expected at the other firm, w > p e j, or else the expected benefit from search is negative and no consumer searches beyond the first firm. Given the optimal search behavior above, retailer i s demand is given by Q i = (1 G(w p e j + p i )) + w p e j +p i p i + G(w p e i + p j )(1 G(w p e i + p i )) + G(p j p i + v)g(v)dv w p e i +p i p i G(p j p i + v)g(v)dv. (2) The first term is the demand from consumers who visit retailer i and buy outright because their utility draw is smaller than the threshold w p e j + p i. The first integral is the demand from those consumers who first visit retailer i, decide to visit retailer j, but come back and buy from retailer i. The third term is the demand from consumers who first visit retailer j, draw a value v j that is smaller than the threshold value w p e i + p j, decide to visit retailer i, encounter a value v i that is higher than w p e i + p i and buy at retailer i. Finally, the second integral is similar to the third term, but accounts for those consumers who first visit retailer j, decide to search retailer i, also draw a relatively low value at retailer i, but still buy at retailer i as that price/product offer is considered to be better than that offered by retailer j. This expression is different from the standard expression in the literature following Wolinsky (1986) in several respects. It allows retailer i, whose demand is being computed, to set a price p i, which may differ from all of the following: (i) the price of its rival, p j, (ii) the belief consumers who visit firm 2 hold about its price, p e i, and (iii) the belief consumers who visit it hold about firm 2 s price, p e j. When looking for the equilibrium with passive beliefs, the Wolinsky literature sets all p j, p e j and p e i equal to the candidate equilibrium price p. As our aim is to study symmetric beliefs, we must allow p e j to 7

potentially change with p i. Furthermore, we will study two models where firms common marginal cost is known to them, but unknown to consumers. Because of this, our demand equation has to allow for the possibility that firm i is aware that its price differs from retailer j s price (because retailer i is planning to deviate), which in turn differs from the belief consumers who visit it hold about retailer i s price, and also consumers who visit retailer i have beliefs about the price of retailer j that are different from all the previous prices. The demand expression for firm i is illustrated in Figure 1. Area A corresponds to the first term in (2) which refers to consumers who arrive at firm i first, draw v i above w p i + p e j and buy outright. Area B corresponds to consumers who arrive at i first but continue to search because their utility draw v i is smaller than w p i + p e j and then come back to buy because their utility draw at firm j is even worse (the second term in (2)). Areas C and D correspond to consumers who first visit firm j, draw a utility level v j that is smaller than w p j + p e i, continue to search firm i and purchase there as their utility at firm i is higher than p i and higher than the utility at firm j. Note that consumer beliefs affect who searches, but not whether consumers buy or not. v i v i A C w + p i p e j w + p i p e i B D p i p i v j + p i p j v j + p i p j p i p j p i p j p j w + p j i) ii) p e j v j p j w + p j p e i v j Figure 1: Figures i) and ii) depict consumers who buy from firm i after having made a first visit to firm i and firm j, respectively. 3 Equilibrium Behavior In this section, we characterize the Perfect Bayesian Equilibrium (PBE) of the Wolinsky model for both passive and symmetric beliefs. This is the key ingredient for our subsequent analysis. The analysis with passive beliefs is mostly well known, except for the part where we solve the model for large search cost (or alternatively, for large marginal cost). We show that once search cost exceeds a certain threshold, demand drops discontinuously. 8

The analysis with symmetric beliefs is new, and also features a drop in demand. This drop-in-demand feature of the model is important when we consider vertical relations, because under symmetric beliefs a manufacturer has the incentive and the ability to avoid such a drop. To be precise about the role of beliefs in the Wolinsky model, we formally define the equilibrium notion below. Beliefs about the price to be observed at the next search may depend on the price observed at the first search, so consumers expected utility of searching another retailer may depend on the price observed at the first firm, and we write w(p 1 ). Definition 1 A symmetric perfect Bayesian equilibrium (PBE) of the Wolinsky model is a price p (the same for both firms) and a reservation utility w(p 1 ) such that 1. Each firm i chooses p i = p to maximize its expected profit given the price of the other firm and consumers reservation utility; 2. Consumers follow an optimal search rule given their beliefs and the match value v i and price p i they observe at firm i; 3. Consumers beliefs are updated using Bayes Rule when possible, i.e., whenever they observe p 1 = p at their first search, they believe that the other firm has also set a price p. Out-of-equilibrium beliefs p e 2 of the price set by the firm that is not yet visited are either i. (passive beliefs) such that p e 2 = p if p 1 p, or ii. (symmetric beliefs) such that p e 2 = p 1 if p 1 p. It is important to understand why we focus on the duopoly case. First, focusing on duopoly allows us to simplify the analysis considerably. In particular, with two retailers, a consumer has to form a belief once (prior to the first search), and so her belief-formation is relatively simple. This is not true for three or more retailers, where it may be the case that, after having searched two firms, a consumer observes two different prices and needs to re-form her belief in order to decide whether to search or not. Second, regardless of how subsequent beliefs are formed, the issues we uncover in the duopoly model are still relevant because they concern the first search. In fact, in virtue of the fact consumers are more, and often much more, likely to search once rather than two or more times, what our analysis in this paper is of first-order relevance even in oligopoly models. It is intractable to provide a proof that the equilibrium exists for a general match function G( ). As far as we know none of the papers in the literature based on the Wolinsky model provides such a general existence proof. The existence proof is trivial, however, when G( ) is uniform, where it is also easy to see that the symmetric equilibrium is 9

unique. Moreover, numerical simulations show that the existence and the uniqueness of the symmetric equilibrium are also guaranteed for other frequently used distribution functions, such as the normal and logistic distribution. The propositions below are formulated assuming that the equilibrium exists. 3.1 Passive Beliefs In order to find the symmetric equilibrium of the Wolinsky model with passive beliefs, we use demand from (2) and adapt it to passive beliefs. Let p denote the equilibrium price where it does not exceed w. Then from the perspective of retailer i who intends to deviate to p i p, retailer j charges the equilibrium price, and consumers who first visit retailer i or retailer j also expect the other retailer to charge the equilibrium price, no matter what price they observe at their first visit. Therefore, we set p j = p e j = p e i = p. to: When the equilibrium price satisfies w p, expected demand for retailer i simplifies Q i = (1 G(w p + p i ))(1 + G(w)) + 2 w p +p i p i G(p p i + v)g(v) dv. The first-order condition for firm 1 s profit maximization, along with the equilibrium condition p i = p gives the following pricing rule: p 1 G(p ) 2 = ˆp c + 2 w g(v) p 2 dv + 2G(p )g(p ) + (1 G(w))g(w), (3) where ˆp is defined as the equilibrium price under passive beliefs for low search cost. Note that ˆp is an increasing function of c. As shown by Anderson and Renault (1999), ˆp is also increasing in s when 1 G(v) is logconcave. The equilibrium price is derived under the assumption that all consumers search, which in this setting requires that the benefit from search, equal to w ˆp, is non-negative. For high enough s, however, the solution to (3) exceeds w. This is because w is a decreasing function of s, and so there always exists a sufficiently large s such that ˆp > w. Each firm is then a single-good monopolist that faces demand 1 G(p). The resulting profit (p c)(1 G(p)) is maximized at the monopoly price, denoted by p M, that solves p M = c + 1 G(pM ). (4) g(p M ) The threshold search cost level s 2 (s 1 will be defined shortly) is such that w = ˆp at which point (3) and (4) coincide. Therefore, s 2 solves v p M (v p M )g(v) dv = s 2. (5) Proposition 1 Under passive beliefs, the equilibrium price p of the Wolinsky model is 10

equal to ˆp for s s 2, and is equal to p M for s > s 2. Figure 2 shows that the equilibrium price is continuous in search cost s and strictly increasing for s s 2 and constant for s > s 2. To intuitively understand why for passive beliefs ˆp is equal to p M when ˆp = w it is useful to consider in Figure 1 a marginal upward deviation from p i = ˆp when ˆp is close to w and p e j = ˆp. The first thing to note is that for ˆp close to w and p e j = ˆp, the areas B and D are negligibly small. Moreover, in areas B and D, the number of consumers firm i loses by increasing its price is (almost) proportional to the size of these areas. The second thing to note is that a monopolist considers the value of the demand he loses by increasing price relative to the increased revenue he makes over all consumers that continue to buy. As valuations for product i and j are independent of each other and the losses are proportional to the size for all areas A-D, a firm that under passive beliefs marginally increases its price for ˆp close to w and p e j = ˆp has identical considerations as a monopolist (who only considers area A). More intuitively, as s increases and w approaches ˆp, almost no consumers search and compare prices. Thus, each firm is only concerned with whether or not consumers accept their price, which is exactly the consideration of a monopolist. Note that for s higher than the threshold value s 2, equilibrium demand falls discontinuously. For s > s 2, consumers only buy when their match value at the first firm exceeds ˆp, so that total demand equals 1 G(p). For lower s, consumers will search on and total demand equals 1 G(p) 2, so the equilibrium demand (and thus retailers profits) is discontinuous at s = s 2. Figure 4 shows the equilibrium level of total demand as a function of search cost. One may wonder why retailers cannot prevent demand from decreasing discontinuously at s = s 2 by collectively abstaining from increasing their prices above w? The reason is that the demand drop is a pure externality - demand at firm i drops because consumers who first visit firm j do not search, but firm i has no way to induce those consumers to search. In other words, when s > s 2, even though both firms would prefer to collectively charge p 1 = p 2 = w, individually they have an incentive to increase their price to a level above w. The equilibrium for s > s 2, where firms charge the monopoly price and no consumer searches beyond the first firm, closely resembles the Diamond Paradox, but in the current setting, unlike Diamond s, the first search is free and so the market does not fully break down (see Appendix C for the version of our model with costly first search). 3.2 Symmetric Beliefs Having solved the model for passive beliefs, we now proceed to constructing a PBE with symmetric beliefs. Let p denote the equilibrium price under symmetric beliefs when it 11

does not exceed w. In order to find the expected demand for retailer i at p i p, we note that with symmetric beliefs p e i = p j = p if only firm i deviates. This follows from the supposition that retailer j does not deviate from the equilibrium price p and hence consumers who visit it first expect firm i to also stick to its equilibrium strategy. Unlike the model with passive beliefs, p e j p but rather p e j = p i. Thus, under symmetric beliefs expected demand for retailer i charging p i, is given by Q i = G(w p + p i ) + w p i G( p p i + v)g(v)dv + For small s, the equilibrium price is then given by w p+pi p i G( p p i + v)g(v)dv. (6) 1 G( p) 2 p = c + 2 w p g(v)2 dv + 2G( p)g( p). (7) Recall that (7) is derived under the assumption that p < w. As w is decreasing in s, for large enough s we have p = w and p = c + 1 G(p)2 2g(p)G(p). This price turns out to be the profit-maximizing price for a monopolist selling both products. Such a monopolist would set p 1 = p 2 = p and maximize (1 G(p) 2 )(p c), which is maximized at p JM that solves p JM = c + 1 G(pJM ) 2 2g(p JM )G(p JM ). (8) The joint profit maximizing price p JM is always higher than the single-product monopoly price p M because joint profit maximization takes into account demand externalities between the two products. Therefore, for a range of search costs, the two firms set prices that are higher than the single-good monopoly price and as high as the price of a multigood monopolist. It is surprising that a collusive outcome is achieved via consumer beliefs, while firms act independently, given these beliefs. To explain this joint profit maximizing result, it is useful to return to Figure 1 and consider a marginal upward deviation from p i = ˆp when ˆp is close to w and p e j = p i (and p e i = ˆp). With symmetric beliefs, firm i does not trigger additional searches (the lower bound of area A is w which is independent of p i ). Thus, an increase in p i only results in the shrinking of area B, which when this area is very small is roughly equivalent to losing only those consumers whose valuations for i are smaller than p i, and at the same time whose valuations for j are even lower. Thus firm i only loses marginal consumers who prefer product i to j. This is exactly how a joint profit maximizing monopolist acts - when increasing p i it only loses consumers whose valuation for good i is higher than for 12

good j: other consumers simply switch to the other good. Thus, for sufficiently large s, firms act as if they are jointly maximizing profits. The highest search cost for which p is the equilibrium price is such that w = p JM. Denote this level of search cost by s 1. It solves v (v p JM )g(v) dv = s 1. (9) p JM Then for s s 1, the equilibrium price is given by p. As p JM > p M, it is clear that s 1 < s 2. Now consider equilibrium prices for s > s 1. Note that in this case w < p. This means that whatever the symmetric equilibrium price, it cannot be lower than w because in that case each firm wants to deviate to a higher price. Can it be the case that firms charge equilibrium prices above w? In this case, consumers do not search and firm 1 s demand is the monopoly demand 1 G(p) and profits are maximized at p M. Depending on whether w is lower or higher than p M there are two cases. First, if s 1 < s s 2 (and w p M ), firms will never want to charge a price higher than w as this reduces profits even further. So the only possible candidate for the equilibrium price is p 1 = p 2 = w. For this interval of search cost the price is decreasing in s. Second, if s > s 2 (and w < p M ), firms cannot stop themselves from deviating to prices higher than w for the same reasons as explained under passive beliefs, even though they suffer from a discrete drop in demand. Thus, the equilibrium price is p M and consumers stop searching. The following proposition summarizes. Proposition 2 Under symmetric beliefs, the retailers equilibrium price for the Wolinsky model is equal to p for s < s 1, is equal to w for s 1 s s 2, and is equal to p M for s 2 < s. Figure 2 illustrates how the equilibrium price varies with search cost in the Wolinsky model with symmetric beliefs (it also shows the price for passive beliefs for comparison). For low search cost the price first rises in s, until the joint monopoly price, and then falls until it is equal to the monopoly price. For higher s, the equilibrium price is constant and equal to the monopoly price. 7 It is easy to establish that for s (0, s 2 ), the equilibrium price under symmetric beliefs, is strictly higher than under passive beliefs and non-monotonic in s. Furthermore, for s (s 1, s 2 ), where price is decreasing in s, it does not even depend on firms marginal cost c, and is fully determined by the reservation utility w. Figure 4 also shows how 7 Unlike the same phenomenon in Janssen et al. (2005), in our model the non-monotonicity of the equilibrium price in search cost does not depend on whether or not the first search is costly. With costly first visits, the monopoly pricing for s > s 2 would disappear as the market breaks down (as in the Diamond Paradox), but the part of the equilibrium construction where s 1 < s < s 2 would not be affected. 13

0.7 w p JM p p M ˆp 0.4 0 s 1 0.1 s 2 0.2 Figure 2: Price in the Wolinsky model with passive (light red) and symmetric (dark blue) beliefs as a function of s for G( ) U(0, 1) and c = 0. s equilibrium total demand depends on the search cost under symmetric beliefs. It clearly shows the drop in demand when search cost increases beyond s 2. Having characterized the equilibrium in terms of search cost s, we now turn to the characterization in terms of c. For this purpose, fix a search cost s, and consider what happens for various levels of c. Let p JM (c) and p M (c) be defined as before, where c in brackets highlights the dependence on c. It is clear that p JM (c) and p M (c) are increasing in c, and reach v when c is sufficiently high, at least in the limit. Then, for any given s, define c 1 and c 2 as solutions to p JM (c 1 ) = w and p M (c 2 ) = w, respectively. 8 These thresholds can be defined in terms of w from (4) and (8): c 1 = w 1 G(w)2 2g(w)G(w), (10) c 2 = w 1 G(w). (11) g(w) For any c we have p JM (c) > p M (c) so that c 1 < c 2. We can now state a counterpart to Proposition 2 in terms of c. Proposition 3 Under symmetric beliefs, the Wolinsky model has a unique equilibrium where the downstream price is p for c < c 1, is equal to w for c 1 c c 2, and is equal to p M for c > c 2. Note that the first two regions of marginal cost in the Proposition may be empty for sufficiently high s if we impose the natural restriction that c 0. When s is sufficiently low, so that c 1 > 0, we have the following comparative statics with respect to c. When 8 Note that both values can be negative when w is sufficiently small. We keep this in mind in what follows. 14

c < c 1 price equals p(c), which is increasing in c. For c [c 1, c 2 ] price equals w, which is independent of c. The flat part stems from the fact that for c [c 1, c 2 ], the search cost s is in the interval [s 1, s 2 ]. Finally, when c > c 2, price equals p M (c) and is again increasing in c. See Figure 3 for an illustration. For comparison, the figure also shows equilibrium prices under passive beliefs. Figure 5 shows how equilibrium total demand depends on the firm s marginal cost under symmetric beliefs. It clearly shows the drop in demand from 1 G(w) 2 to 1 G(w) when the search cost increases beyond c 2. The figure also displays the dependence under passive beliefs. 0.9 ˆq q q M 0.4 s s 1 0.1 s 2 0.2 Figure 3: The equilibrium demand in the Wolinsky model with symmetric (dark blue) and passive (light red) beliefs as a function of s for G( ) U(0, 1). 0.75 p M (c) w p(c) ˆp(c) 0.4 0 c 1 0.25 c 2 0.5 c Figure 4: The equilibrium price in the Wolinsky model with symmetric (dark blue) and passive (light red) beliefs as a function of c for G( ) U(0, 1). 15

0.8 ˆq q 0.5 q M 0.2 c 0 c 1 0.25 c 2 0.5 Figure 5: The equilibrium demand in the Wolinsky model with symmetric (blue) and passive (red) beliefs as a function of c for G( ) U(0, 1). 4 Asymmetric Information about Common Cost In this section, we study the environment where firms have a common cost that is known to them, but unknown (and uncertain) to consumers. 9 Let firms common marginal cost be drawn from a continuous distribution F ( ) with support [c, c]. Firms observe the cost before posting prices, but consumers do not. We concentrate on symmetric equilibria of this model. We first define P as the reach of prices in equilibrium. Thus P is the set of prices that are observed along the equilibrium path. Because equilibrium is symmetric, if a consumer arrives at firm i and observes p i P, she has to believe that firm j also charges p. That means that symmetric beliefs are implied on the equilibrium path and for all prices in P beliefs have to be symmetric! For all out-of-equilibrium p i P, consumers may have different beliefs. For simplicity, however, we concentrate on equilibria where beliefs are also symmetric for prices p i P. In this asymmetric information model where retailers cost is randomly determined, retailers (symmetric) strategy is given by a function p(c), whereas the search strategy is characterized by w (p 1 ). Formally, a symmetric equilibrium for this model is then given as follows. Definition 2 A symmetric perfect Bayesian equilibrium in continuous strategies of the search model with asymmetric information about cost is a continuous pricing strategy p (c) with reach P and a reservation utility w (p 1 ) such that 1. for every cost realization c, each firm i chooses p i (c) = p (c) to maximize its expected profit given the pricing strategy p (c) of the other firm and consumers reservation utility w(p 1 ); 9 The effect of common cost uncertainty has been studied in the context of the Stahl (1989) model of sequential price search with homogeneous goods (see, e.g., Dana (1994), Tappata (2009) and Janssen et al. (2011)), but not in the differentiated goods sequential search model based on Wolinsky (1986), where the equilibrium price follows a pure strategy. 16

2. Consumers follow an optimal search rule given their beliefs and the match value v i and the price p i they observe at firm i; 3. Consumers beliefs are updated using Bayes Rule when possible, i.e., whenever they observe p 1 P at their first search, they believe that the other firm has also set a price p e 2 = p 1. Out-of-equilibrium beliefs p e 2 of the price set by the firm that is not yet visited are symmetric, i.e., p e 2 = p 1 if p 1 / P. Given that beliefs are symmetric, the model can be easily solved by using Figure 4 and noting that for any realization of c, the equilibrium price is given by the blue curve. Then, depending on the relationship between c 1 and c 2 on the one hand, and c and c on the other, we have one of the following equilibria: Proposition 4 For any given s, depending on the range of cost uncertainty [c, c] the equilibrium of the asymmetric information model takes one of the following forms: (a) If c c 1, the equilibrium is fully separating and the equilibrium price is p(c) for cost realization c. (b) If c < c 1 < c, the equilibrium is partially separating and the equilibrium price is p(c) for c < c 1 and w for c 2 c c 1. If, in addition, c > c 2, then for c > c 2, the equilibrium price is p M (c). (c) If c 1 c < c c 2, the equilibrium is pooling and the equilibrium price is w for all c; (d) If c < c 2 < c, the equilibrium is partially separating and the equilibrium prices is w for c 1 c c 2, and p M (c) for c 2 < c. If, in addition, c < c 1, the equilibrium price is p(c) for c < c 1. (e) If c c 2 the equilibrium is fully separating and the equilibrium price is p M (c). While the Proposition has many cases, each of them can be understood by placing the interval [c, c] into Figure 4. As the blue curve has increasing and flat parts, the equilibrium can be fully separating, fully pooling or partially separating, depending on where c and c fall relative to c 1 and c 2. For instance, if the exogenous cost uncertainty is small (c is close to c), the equilibrium is likely to be fully separating or fully pooling, whereas if the range of cost uncertainty is large, it will be partially separating. It is interesting to relate some aspects of this equilibrium characterization to two independent literatures: the literature on sticky prices and the literature on sales. Starting with the seminal contributions by Sheshinski and Weiss (1977), Akerlof and Yellen (1985a,b) and Mankiw (1985), there is a large literature that explains why firms do not 17

adjust prices following cost shocks assuming an exogenous cost of price adjustment, the menu cost. Our model generates price stickiness without assuming menu cost. Instead, prices are sticky because, for a range of marginal costs, firms find it optimal to set prices equal to the consumer s reservation utility that is independent of marginal cost. Sherman and Weiss (2014) empirically find that retailers in the Shuk Mahane Yehuda market in Jerusalem do not react to cost changes. They explain this finding with a dynamic model where consumers are not informed about the cost and do not adjust their expectations about prices charged by other retailers. In this world, if retailers were to increase their prices consumers would walk away to the next store. Compared to their paper, we have a static model with product differentiation and a continuum of cost states. Cabral and Fishman (2012) have also proposed a search theoretic foundation for price stickiness. Their framework relies, however, on some stickiness in retailers cost and they show that consumer search may lead to retail prices that are even stickier. Our results in Proposition 4 do not rely in any way on stickiness of retailers cost. The result that prices are fully rigid does depend, however, on the assumption that all consumers have identical search costs. If search costs are heterogeneous (see, also, Moraga, Sandor and Wildenbeest (2014)), but the search cost distribution is concentrated around a certain value, prices will be almost sticky, and we would obtain a search theoretic explanation for incomplete cost pass-through (Weyl and Fabinger (2013)). Following the seminal article by Varian (1980), the literature on sales provides an explanation for the common observation that retailers often follow a pricing strategy for their products that is characterized by a regular price most of the time and a sales price at random moments in time, where the discount given on the regular price is also subject to large variations (see, e.g., Narasimhan (1988)). This literature explains this phenomenon by an asymmetric mixed strategy equilibrium with a mass point for the regular price and a continuous price distribution for the sales prices. From the viewpoint of the consumers (or the empirical economist who does not have data on cost, but only on retail prices) the equilibrium under b) in the Proposition can also be viewed as an equilibrium with sales. For many cost levels, firms (regular) price is independent of cost, but when cost becomes sufficiently low, the retailers give a discount. When cost is unobserved by the empirical economist, the firm acts as if it is giving a random price promotion, but in fact (in our model) the sale arises as the realization of a pure strategy that is symmetric across firms. Empirically, Pesendorfer (2002) finds evidence for the fact that a firm has a regular price and then provides discounts at irregular points in time. Note, however, that in line with (d) in Proposition 4, a firm may also have a regular price with irregular price movements going up and down. This is the pricing pattern that is documented in Hosken and Reiffen (2004). Before we proceed to the comparative statics of this equilibrium, note that there is no equilibrium where beliefs are passive everywhere. This is because with passive beliefs, 18

price is increasing in c for all c (see the red curve in Figure 4). Therefore the interval [c, c] maps into some interval P = [p(c), p(c)], but for all p P, beliefs have to be symmetric, which contradicts the initial supposition that beliefs are passive. While there may exist equilibria where beliefs are symmetric for p P, and passive for p P, we shall not pursue these further. Comparative statics The comparative statics of this model with respect to search cost depends on which of the five cases described in Proposition 4 applies. There are two fully separating cases, (a) and (e). In (a), the distribution of prices shifts to the right as s increases, while in (e), prices are independent of s. In all partially separating and pooling equilibria, for a range of marginal cost values, price equals w, and because w is decreasing in s, so is the price for any corresponding c. As an example, consider case (b) where c < c 1 < c, which has a frequently observed structure of a regular price w for cost realizations c c 1 and random discounts for cost realizations c < c 1 where the price is p(c) < w. In this case, prices are distributed continuously on the interval [ p(c), w) with a CDF given by F ( p 1 (p)), where p 1 ( ) is the inverse of p(c), and there is a mass point equal to 1 F (c 1 ) at w. Now consider a marginal increase in s. It shifts the distribution of prices below w to the right, as per increase in p(c) for every c, but w shifts to the left, while the mass at w increases. Thus, an increase in search cost leads to a lower and more frequently charged regular price (w), but shallower and less frequent discounts. Whether the average price increases or decreases is ambiguous, and depends on G( ) and F ( ). Another way to think about comparative statics is to fix a narrow interval [c, c] (in the sense that all five cases will appear for some s), and consider what happens as s changes. When s is low, equilibrium falls into case (a) and as s increases, the price increases. As s increases further, equilibrium transits from (a) to (e), where the price becomes independent of s. In (b) and (d) average price may be falling or increasing, but in (c) where the price equals w it is falling. 5 Vertical Industry Structure We now turn to the environment where retailers cost is (partially) determined by a manufacturer and consider the impact of consumer beliefs on prices and upstream and downstream profits. As consumers do not know the retail prices (and have to pay a search cost to observe them), it is realistic to also assume that consumers do not observe the wholesale price. 10 Note that in this setting the final retail price consumers observe 10 Janssen and Shelegia (2014) focus on the comparison of markets where consumers observe the wholesale price and ones where they do not observe the wholesale price in the context of the Stahl (1989) 19